Volume 9, Issue 2
Approximate Riemann Solvers and Robust High-order Finite Volume Schemes for Multi-dimensional Ideal MHD Equations

Franz Georg Fuchs, Andrew D. McMurry, Siddhartha Mishra, Nils Henrik Risebro & Knut Waagan

Commun. Comput. Phys., 9 (2011), pp. 324-362.

Published online: 2011-09

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  • Abstract

We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semiconservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshes.

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@Article{CiCP-9-324, author = {Franz Georg Fuchs, Andrew D. McMurry, Siddhartha Mishra, Nils Henrik Risebro and Knut Waagan}, title = {Approximate Riemann Solvers and Robust High-order Finite Volume Schemes for Multi-dimensional Ideal MHD Equations}, journal = {Communications in Computational Physics}, year = {2011}, volume = {9}, number = {2}, pages = {324--362}, abstract = {

We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semiconservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.171109.070510a}, url = {http://global-sci.org/intro/article_detail/cicp/7502.html} }
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